Harmonic-oscillator functions table

The DF spectra of wurtzite-structure ZnO within the VIS-to-VUV spectral region contain CP structures, which can be assigned to band-gap-related electronic band-to-band transitions Eq with a = A, B,C and to above-band-gap band-to-band transitions E13 with (3 = 1,..., 7. The F -related structures can be described by lineshape functions of the 3DMo-type (3.9 and 3.10), the CP structures with (3 = 3,4 by lineshape functions of the 2DMo-type (3.11), and the CP structures with (3=1,2,5,6,7 can be described by Lorentzian-damped harmonic oscillator functions (3.13). The CP structures Eq are supplemented by discrete (3.14) and continuum (3.16) excitonic contributions. Tables 3.9 and 3.10 summarize typical parameters of the CPs Eq and E, respectively, of ZnO [15]. 

By force constants, we refer to derivatives of the electronic energy with respect to internal geometrical parameters of a molecule (Table 1). llie harmonic force constants are the second derivatives evaluated at an equilibrium structure, while higher derivatives may be put in the category of anharmonic force constants. From harmonic constants, harmonic frequencies are immediately obtained, and, typically, the harmonic frequencies are at least accurate enough to be used in making zero-point correaions to stabilities. Given harmonic constants and the lowest one or two orders of anharmonic force constants (third and fourth derivatives), transition frequencies of small polyatomics can often be extracted. Usually, this involves a perturbative treatment of the anharmonic parts of the potential in a produa basis of harmonic oscillator functions. 

Under most circumstances the equations given in Table 10.4 accurately calculate the thermodynamic properties of the ideal gas. The most serious approximations involve the replacement of the summation with an integral [equations (10.94) and (10.95)] in calculating the partition function for the rigid rotator, and the approximation that the rotational and vibrational partition functions for a gas can be represented by those for a rigid rotator and harmonic oscillator. In general, the errors introduced by these approximations are most serious for the diatomic molecule." Fortunately, it is for the diatomic molecule that corrections are most easily calculated. It is also for these molecules that spectroscopic information is often available to make the corrections for anharmonicity and nonrigid rotator effects. We will summarize the relationships... 

By starting with this partition function and going through considerable mathematical manipulation, one arrives at the following equations for calculating the corrections to the rigid rotator and harmonic oscillator values calculated from Table 10,4, U... 

Some of the more useful matrix elements for the harmonic oscillator are presented in the following table. They are given as functions of the dimensionless quantities = Inx vmfh and a = 2sf /tv, as defined in Section 6.2. 

Table 4.1 Partition functions evaluated in the rigid rotor harmonic oscillator approximation... 

In Section 4.8, Equations 4.78,4.79 and Table 4.1 develop the connections between the harmonic oscillator rigid rotor partition function and isotope chemistry as expressed by the reduced partition function ratio, RPFR = (s/s ) f. RPFR is defined in Equation 4.79 as the product over oscillators of ratios of the function [u exp(—u/2)/ (1 - exp(u))]... 

Expressions for the partition function can be obtained for each type of energy level in an atom or molecule. These relationships can then be used to derive equations for calculating the thermodynamic functions of an ideal gas. Table 11.4 or Table A6.1 in Appendix 6 summarize the equations for calculating the translational, rotational, and vibrational contributions to the thermodynamic functions, assuming the molecule is a rigid rotator and harmonic oscillator.yy Moments of inertia and fundamental vibrational frequencies for a number of molecules are given in Tables A6.2 to A6.4 of Appendix 6. From these values, the thermodynamic functions can be calculated with the aid of Table 11.4. 

Table A6.1 summarizes the equations needed to calculate the contributions to the thermodynamic functions of an ideal gas arising from the translational, rotational, and vibrational degrees of freedom. The equations assume that the rigid rotator and harmonic oscillator approximations are valid.

The thermodynamic functions were estimated from those in the present table for HgS(g) (6 ) by adding those for DgS(g) and subtracting those for HgS(g), where both the added and subtracted functions were generated using the rigid-rotor harmonic oscillator approximation. In this calculation the molecular constants for DgS were taken from reference (2). 

The thermodynamic functions were taken from the JANAF table for H2S(g) dated Dec. 31, 1965 (1 ). These in turn were taken from Gordon (8 ) except below 298 K were they were calculated by the rigid-rotor, harmonic-oscillator approximation. Gordon had calculated from 298 K to 6000 K by a method which takes into account second-order corrections for vibrational anharmonicity, vibration-rotation interaction, and centrifugal stretching. The spectroscopic constants used were taken from Allen and Plyler (9). 

We simply list the solutions, which you can verify by substituting them into the Schrodinger equation. The first four wave functions for the quantum harmonic oscillator are listed in Table 4.2 and plotted in Figure 4.31. The energy levels of the harmonic oscillator are given by... 

Fairly good agreement exists between the calculated value of 1682 cm-1 and the experimental value of 1650 cm1. Direct correlation does not exist because Hooke s law assumes that the vibrational system is an ideal harmonic oscillator and, as mentioned before, the vibrational frequency for a single chemical moiety in a polyatomic molecule corresponds to the vibrations from a group of atoms. Nonetheless, based on the Hooke s law approximation, numerous correlation tables have been generated that allow one to estimate the characteristic absorption frequency of a specific functionality (13). It becomes readily apparent how IR spectroscopy can be used to identify a molecular entity, and subsequently physically characterize a sample or perform quantitative analysis. 

Figure 8. Vibration-rotation contributions to the entropy for H20 as a function of temperature. The lower curve (O, solid line) shows the entropy predicted by the classical rigid rotator-quantum harmonic oscillator (RR/HO) model, the intermediate curve ( , dashed line) shows data taken from the JANAF Thermochemical Tables, and the upper curve (0> solid line) is obtained from differentiating the fourth-order fit to the AOSS-U free energy calculations shown in Figure 6. Note that (as expected) all three curves coincide at low temperatures and that the JANAF and AOSS-U results are in dose agreement throughout the entire temperature range.

Using the rigid-rotor harmonic-oscillator approximation on the basis of molecular constants and the enthalpies of formation, the thermodynamic functions C°p, S°, — G° —H°o)/T, H° — H°o, and the properties of formation Af<7°, and log K°(to 1500 K in the ideal gas state at a pressure of 1 bar, were calculated at 298.15 K and are given in Table 9 <1992MI121, 1995MI1351>. Unfortunately, no experimental or theoretical data are available for comparison. From the equation log i = 30.25 - 3.38 x /p t, derived from known reactivities (log k) and ionization potential (fpot) of cyclohexane, cyclohexanone, 1,4-cyclohexadiene, cyclohexene, 1,4-dioxane, and piperidine, the ionization potential of 2,4,6-trimethyl-l,3,5-trioxane was calculated to be 8.95 eV <1987DOK1411>. 

Computer Program for the Numerov Method. Table 4.1 contains a BASIC computer program that applies the Numerov method to the harmonic-oscillator Schrodinger equation. The character in the names of variables makes these variables double precision. M is the number of intervals between and and equals (x,niax r,o)/ S- Lines 55 and 75 contain two times the potential-energy function, which must be modified if the problem is not the harmonic-oscillator. If there is a node between two successive values of x then the values at these two points will have opposite signs (see Problem 4.43) and statement 90 will increase the nodes counter NN by 1. 

Thermodynamic functions of PHJ [51] in the ideal gas state, calculated in the rigid rotor-harmonic oscillator approximation, are presented in Table 19. They are based on the fundamental frequencies derived from the vibrational spectra of phosphonium halides (see above) and the calculated bond length [10] r=1.382 A [50]. 

The coirrses of the output functions caused by the generation of the sinusoidal input function for proportional, integrating and inertial first-order objects are also presented in Table 2.1. It is seen that, for a proportional object, only the value of the sinusoidal (harmonic) oscillation frequency changes. For an integral object, the sinusoidal input function is transformed by the object to a cosinusoidal function. The amplitude of the output function is then inversely proportional to the frequency of the... 

The translation-invariant decomposition (9.38) was first written by Post [95] and was rediscovered independently in refs. [85,86], The result (9.39) clearly constitutes an improvement with respect to the previous inequality (9.18) because the constituent mass in is decreased by a factor 3/4, and therefore the energy E. is algebraically increased. For an attractive power-law potential e(/3)r, this provides a factor (4/3). A numerical comparison is shown in table 9.1, where are listed the naive lower limit (9.18), the improved lower limit (9.39), the exact energy obtained by a hyperspherical expansion, and the variational bound derived from a Gaussian trial wave function. It is worth noticing that the new lower limit (9.39) becomes exact in the case of the harmonic oscillator. This is true for an arbitrary number A of bosons and the harmonic oscillator is the only potential for which the inequality is saturated. A beautiful proof of this property has been given by Wu and is included in ref. [ ]. 

Table 1 Equations for the translational, rigid-rotational, and harmonic oscillator contributions to the thermodynamic functions of molecules in the ideal-gas state and at the standard pressure 101325 Pa...

Thus the vibrational contribution to the partition function can be written exactly in closed form. The resulting practical equations for the thermodynamic functions are given in Table 1. The total vibrational contributions will be the sum of the contributions from each of the vibrations a vibration Vi of degeneracy dt will contribute di times the contribution of that Several tabulations of the harmonic oscillator or Einstein functions are available the most reliable are those giving the values in dimensionless form and the most convenient for practical use are those of ref. 13. 

The selected molecular constants and electronic state excitation energies were used to calculate the thermodynamic functions in the rigid rotator-harmonic oscillator approximation over the temperature range 298.15-3000 K at standard pressure. These functions are gathered in Table A2 in the form of the coefficients of the polynomial (see Appendix). 

We have adopted a vialue of AHS(CF2,g,298) = -44.6 kcal/mol from the data of Modica and LeGraff (16,17) and of Carlson (19). This yields values of the equilibrium constant for reaction t ) with in a factor of two of those calculated from the data of Farber et (21), which is certainly within the accuracy of both the experiment and the limits of the rigid-rotor, harmonic oscillator approximation at 2000 to 2500 K ( ). The physical and thermochemical data selected here are sumnarized in Table II and the ideal gas thermodynamic functions calculated to 1500 K from these data are summarized in Table III. 

---